Related papers: On the growth of real functions and their derivati…

For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^\gamma <\infty$, $\gamma>4$ implies the almost everywhere convergence of $\sum_{k=1}^\infty c_k…

Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved…

We prove a functional extension of an exponential inequality originally proposed by Bin Zhao and proved by Xiaosheng Mou. The main result asserts that if $\alpha_1\leq \cdots\leq \alpha_n$ and $\sum_{k=1}^n \alpha_k=0$, then \[ \sum_{k=1}^n…

We prove that if $f(x) = \sum_{k=0}^\infty a_k x^k,$ $a_k >0, $ is an entire function such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$ is non-decreasing and $\frac{a_1^2}{a_{0}a_{2}} \geq 2\sqrt[3]{2},$…

Let $G(k)=\int_0^1g(x)e^{kx}dx$, $g\in L^1(0,1)$. The main result of this paper is the following theorem. {\bf Theorem}. {\it If $\limsup_{k\to +\infty}|G(k)|<\infty$, then $g=0$. There exists $g\not\equiv 0$, $g\in L^1(0,1)$, such that…

Let $f(x)$ be a real function which has $(n+1)$-th derivative on an interval $[a, b]$. For any point $x_0\in (a, b)$ and any integer $0\leq k\leq n$, denote by $S_{k,x_0}(x)$ the $k$-th truncation of the Taylor expansion of $f(x)$ at $x_0$,…

For a real-valued non-negative and log-concave function we introduce a notion of difference function; the difference function represents a functional analog on the difference body of a convex body. We prove a sharp inequality which bounds…

Let $X=\{X_n: n\in\mathbb{N}\}$ be a long memory linear process in which the coefficients are regularly varying and innovations are independent and identically distributed and belong to the domain of attraction of an $\alpha$-stable law…

In this paper, we prove the following result. Let $\alpha$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$ \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x) = 0 , \;\; x \in \mathbb{R}^n ,\\…

Let $f: \mathbb{N}^2 \mapsto \mathbb{C}$ be an arithmetic function of two variables. We study the existence of the limit: \[\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 , n_2 \le x} f (n_1, n_2) \] where $k$ is a…

We show that there is always a uniformly antisymmetric f:A-> {0,1} if A subset R is countable. We prove that the continuum hypothesis is equivalent to the statement that there is an f:R-> omega with |S_x| <= 1 for every x in R. If the…

Fix an integer $h \geq 2$, and let $b_1, \ldots, b_h$ be (not necessarily distinct) positive integers with $\gcd(b_1, \ldots, b_h) = 1$. For any subset $A \subseteq \mathbb{N}$, let $r_A(n)$ denote the number of solutions $(k_1, \ldots,…

We prove that conformable ``fractional" differentiability of a function $f:[0,\infty[\,\longrightarrow \mathbb{R}$ is nothing else than the classical differentiability. More precisely, the conformable $\alpha$-derivative of $f$ at some…

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on $\mathbb{R}^n$ with an explicit expression for the constant. Namely, we show that for all $0<s<\frac{n}{2}$ and $a>0$ we have the…

Let $f$ be a nonnegative function of class $C^k$ ($k \geq 2$) such that $f^{(k)}$ is H\''older continuous with exponent $\alpha$ in $(0,1]$. If $f'(x) = \cdots = f^{(k)}(x) = 0$ when $f(x) = 0$, we show that $f^{\mu}$ is differentiable for…

We establish the linear independence of time-frequency translates for functions $f$ having one sided decay $\lim_{x\to \infty} |f(x)| e^{cx \log x} = 0$ for all $c>0$. We also prove such results for functions with faster than exponential…

Let $\alpha\in\,]0,1[$. We prove that the existence of the conformable fractional derivative $T_{\alpha}f$ of a function $f:[0,\infty[\,\longrightarrow \mathbb{R}$ introduced by Khalil et al. in [R. Khalil, M. Al Horani, A. Yousef, M.…

For irrational $\theta$ and 1-periodic function $f$ we consider sums $\sum_0^{Q-1}f(k\theta+\varphi)$ where $\varphi \in \mathbb R$. Sidorov proved that if $f$ is absolutely continuous function, then $\liminf_{Q \to \infty}…

We prove that for any real polynomial $f(x) \in\mathbb{R} [x]$ the set $$ \{\alpha \in \mathbb{R}: \liminf_{n\to \infty} n\log n ||\alpha f(n)|| >0\} $$ has positive Hausdorff dimension. Here $||\xi ||$ means the distance from $\xi $ to the…

In this short note, we establish the following result: Let $f:[0,+\infty[\to [0,+\infty[$, $\alpha:[0,1]\to ]0,+\infty[$ be two continuous functions, with $f(0)=0$. Assume that, for some $a>0$, the function $\xi\to…